Integration Worksheet2 Solution1





In this page integration worksheet2 solution1 we are going to see solution of some practice question from the worksheet of integration.

Question 1

(i) Integrate the following with respect to x , x⁴

Solution:

The given question exactly matches the formula 

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

Now we are going to integrate the given question by using this formula in the question instead of n we have 4

So we get,     

∫ x⁴ dx  = x^(4 + 1)/(4 + 1)

            = (x^5/5) + C


(ii) Integrate the following with respect to x ,   (x + 3) ⁵

Solution:

The given question exactly matches the formula 

∫ (ax + b)ⁿ dx = (1/a) [(ax + b)^⁽ⁿ ⁺ ¹⁾/(n + 1)] + c

Now we are going to integrate the given question by using this formula in the question instead of n we have 5

So we get,     

∫ (x + 3)⁵ dx  = (x + 3)^(5 + 1)/(5 + 1)

            = (x + 3)^6/6 + C

            = (1/6)(x + 3)^6 + C


(iii) Integrate the following with respect to x ,   (3 x + 4) ⁶

Solution:

The given question exactly matches the formula 

∫ (ax + b)ⁿ dx = (1/a) [(ax + b)^⁽ⁿ ⁺ ¹⁾/(n + 1)] + c

Now we are going to integrate the given question by using this formula in the question instead of "n" we have 6 and instead of "a" we have 3

So we get,     

∫ (3 x + 4) ⁶ dx  = (3 x + 4)^(6 + 1)/(6 + 1)

                         = (1/3)[(3 x + 4)^7/7] + C

                         = (1/21)(3 x + 4)^7 + C


(iv) Integrate the following with respect to x ,   (4 - 3 x) ⁷

Solution:

The given question exactly matches the formula 

∫ (ax + b)ⁿ dx = (1/a) [(ax + b)^⁽ⁿ ⁺ ¹⁾/(n + 1)] + c

Now we are going to integrate the given question by using this formula in the question instead of "n" we have 7 and instead of "a" we have -3

So we get,     

∫  (4 - 3 x) ⁷ dx  = (4 - 3 x)^(7 + 1)/(7 + 1)

                         = (-1/3)[(4 - 3 x)^8/8] + C

                         = (-1/24)(4 - 3 x)^8 + C


(v) Integrate the following with respect to x ,    (L x + m) ⁸

Solution:

The given question exactly matches the formula 

∫ (ax + b)ⁿ dx = (1/a) [(ax + b)^⁽ⁿ ⁺ ¹⁾/(n + 1)] + c

Now we are going to integrate the given question by using this formula in the question instead of "n" we have 8 and instead of "a" we have L

So we get,     

∫ (L x + m)⁸ dx  = (L x + m)^(8 + 1)/(8 + 1)

                         = (1/L)[(L x + m)^9/9] + C

                         = (1/9 L)(L x + m)^8 + C

integration worksheet2 solution1 integration worksheet2 solution1